Abstract

Semidefinite cone-invariant (SCI) systems are defined as a class of linear time-invariant (LTI) systems which possess the spectrahedral cone-invariance property. Such systems have rich structure and include a large class of LTI systems, e.g., positive systems. However, they have not received enough attention from the control community. In this thesis, we make some preliminary attempts to study the theory and application of discrete-time SCI systems.

The first part of this thesis is the study of the theory of SCI systems. We discuss some fundamental properties, including spectrahedral cone-stability, observability and detectability of an SCI system. More importantly, we investigate the SCI realization problem. For an LTI system with nonnegative impulse response, much research has been devoted to studying its positive realization. However, the limitations in the eigenvalue positions of nonnegative matrices suggest that positive systems are not adequately powerful as a modeling tool. Hence we propose the SCI realization of nonnegative impulse responses. This is a novel idea and has not appeared in the literature. We can find SCI realizations for a large class of systems with nonnegative impulse responses, which may not have positive realizations.

The second part of this thesis is the study of linear quadratic (LQ) optimal control problem of discrete-time networked control systems with random input gains. It is shown that the solvability of this LQ optimal control problem depends on the existence of a mean-square stabilizing solution to a modified algebraic Riccati equation (MARE). With the help of theory of SCI systems, we provide a necessary and sufficient condition, which is given directly in terms of the system parameters, to ensure the existence of such a mean-square stabilizing solution. Such a condition is derived for the very first time and it indicates that the common condition of the observability or detectability of certain stochastic system is unnecessary. The other highlight is that we put the problem under a channel/controller co-design framework which differentiates our work from a certain pure stochastic optimal controller design problem. Under this framework, the stabilization issue involved can be analytically solved.